Developments in statistics now allow maximum likelihood estimators for the parameters of Markov random fields (MRFs) to be constructed. We detail the theory required, and present an algorithm that is easily implemented and practical in terms of computation time. We demonstrate this algorithm on three MRF models-the standard Potts model, an inhomogeneous variation of the Potts model, and a long-range interaction model, better adapted to modeling real-world images. We estimate the parameters from a synthetic and a real image, and then resynthesize the models to demonstrate which features of the image have been captured by the model. Segmentations are computed based on the estimated parameters and conclusions drawn.

Confocal laser scanning microscopy is a powerful technique for studying
biological specimens in three dimensions (3D) by optical sectioning. It permits
to visualize images of live specimens non-invasively with a resolution of few
hundred nanometers. Although ubiquitous, there are uncertainties in the
observation process. As the system’s impulse response, or point-spread function
(PSF), is dependent on both the specimen and imaging conditions, it should be
estimated from the observed images in addition to the specimen. This problem is
ill-posed, under-determined. To obtain a solution, it is necessary to insert some
knowledge in the form of a priori and adopt a Bayesian approach. The state of
the art deconvolution and blind deconvolution algorithms are reviewed within a
Bayesian framework. In the first part, we recognize that the diffraction-limited
nature of the objective lens and the intrinsic noise are the primary distortions
that affect specimen images. An alternative minimization (AM) approach
restores the lost frequencies beyond the diffraction limit by using total variation
regularization on the object, and a spatial constraint on the PSF. Additionally,
some methods are proposed to ensure positivity of estimated intensities, to
conserve the object’s flux, and to well handle the regularization parameter.
When imaging thick specimens, the phase of the pupil function due to spherical
aberration (SA) cannot be ignored. It is shown to be dependent on the refractive
index mismatch between the object and the objective immersion medium, and
the depth under the cover slip. The imaging parameters and the object’s original
intensity distribution are recovered by modifying the AM algorithm. Due to
the incoherent nature of the light in fluorescence microscopy, it is possible to
retrieve the phase from the observed intensities by using a model derived from
geometrical optics. This was verified on the simulated data. This method could
also be extended to restore specimens affected by SA. As the PSF is space varying,
a piecewise convolution model is proposed, and the PSF approximated so that,
apart from the specimen, it is sufficient to estimated only one free parameter.